00000323.htm

306 X EXISTENCE THEOREMS

Now the bilinear mapping (X, 7)~» X° Y of jSf(E) x^(E) into
is continuously differentiable (8.1.4); denote by R(t) the continuous linear
mapping U-»A(t) ° Uof JS?(E) into itself. From (5.7.5) it follows at once that

\\R(t)~R(t')\\*:\\A(t)-A(t')\\

hence, if K =R, t^R(t) is regulated ift-+A(t) is regulated. On the other
hand, if / -> ^((f) is difFerentiable, so is / -> ^(0, and its derivative at the point
/ (identified (Section 8.4) to an element of <£(E)) is the mapping U-+ A'(t) ° U
((8.1.3) and (8.2.1)); hence if t~+ A'(t) is continuous, so is f-» R'(t}. We can
therefore conclude that if K = C and if / -> A(t) is analytic in I, so is t -> J?(/)
(9.10.1). In any case, we may apply (10.6.4) to the equation (10.8.4.1); let
V(t) be the solution of that equation equal to 7_E for t = s. We have, for any
/el ((8.1. 3) and (8.2.1))

*o) = V(f) • *₀ = -4(0

and furthermore, for t = s, V(s) - x₀ = 7_E • x₀ = x₀ ; it therefore follows from
(10.6.4) applied to (10.8.3) that C(t, s)-x₀= V(t) - x₀ for any JC_Q e E, hence
C(t, s) = V(t) for t e I. This proves that C(t, s) e jSf(E) and that * -> C(/, j) is
the solution of (10.8.4.1) which is equal to /_E for t = s.

Finally, the function /-» C(t, r) - x₀ is the solution of (10.8.3) equal to
C(s₉ r) - X_Q for t = s; hence, by definition

C(t, r)-x₀~ C(t, s) • (C(s₉ r) • x₀) = (C(t, s) ° C(s₉ r)) • x₀

for any x₀ e E, which proves the first relation (1 0.8.4.2) ; as C(t₉ 1) = 7_E , that
relation yields C(t, s) ° C(s, t) = J_E. This shows that C(s₉ 1) is a bfjective linear
mapping of E, whose inverse mapping is C(t₉ s) (hence also belongs to JS?(E)).
With this we reach the end of the proof of (1 0.8.4).

The operator C(t₉ s) is called the resolvent of (10.8.3) (or of (10.6.2)) in I.

(10.8.5) The mapping (s, t) -> C(s₉1) oflxl into jSf(E) is continuous.

We may indeed write C(s₉1) = C(s₉1₀) ° (C(r, /₀))~"\ and the result then
follows from (10.8.4), (5.7.5), and (8.3.2).

The knowledge of C(s₉1) enables one to give the explicit solution of
(10.6.2) taking the value JC_Q for t = t₀:

(10.8.6) The function

ii(0 = C(t, t₀) • x₀ + \(C(t, s) • b(s)) ds

Jto



	306 X EXISTENCE THEOREMS Now the bilinear mapping (X, 7)~» X° Y of jSf(E) x^(E) into is continuously differentiable (8.1.4); denote by R(t) the continuous linear mapping U-»A(t) ° Uof JS?(E) into itself. From (5.7.5) it follows at once that \\R(t)~R(t')\\:\\A(t)-A(t')\\* hence, if K =R, t^R(t) is regulated ift-+A(t) is regulated. On the other hand, if / -> ^((f) is difFerentiable, so is / -> ^(0, and its derivative at the point / (identified (Section 8.4) to an element of <£(E)) is the mapping U-+ A'(t) ° U ((8.1.3) and (8.2.1)); hence if t~+ A'(t) is continuous, so is f-» R'(t}. We can therefore conclude that if K = C and if / -> A(t) is analytic in I, so is t -> J?(/) (9.10.1). In any case, we may apply (10.6.4) to the equation (10.8.4.1); let V(t) be the solution of that equation equal to 7_E for t = s. We have, for any /el ((8.1. 3) and (8.2.1)) o) = V(f)* • *₀ = -4(0

	and furthermore, for t = s, V(s) - x₀ = 7_E • x₀ = x₀ ; it therefore follows from (10.6.4) applied to (10.8.3) that C(t, s)-x₀= V(t) - x₀ for any JC_Q e E, hence C(t, s) = V(t) for t e I. This proves that C(t, s) e jSf(E) and that * -> C(/, j) is the solution of (10.8.4.1) which is equal to /_E for t = s. Finally, the function /-» C(t, r) - x₀ is the solution of (10.8.3) equal to C(s₉ r) - X_Q for t = s; hence, by definition C(t, r)-x₀~ C(t, s) • (C(s₉ r) • x₀) = (C(t, s) ° C(s₉ r)) • x₀ for any x₀ e E, which proves the first relation (1 0.8.4.2) ; as C(t₉ 1) = 7_E , that relation yields C(t, s) ° C(s, t) = J_E. This shows that C(s₉ 1) is a bfjective linear mapping of E, whose inverse mapping is C(t₉ s) (hence also belongs to JS?(E)). With this we reach the end of the proof of (1 0.8.4). The operator C(t₉ s) is called the resolvent of (10.8.3) (or of (10.6.2)) in I.

	(10.8.5) The mapping (s, t) -> C(s₉1) oflxl into jSf(E) is continuous. We may indeed write C(s₉1) = C(s₉1₀) ° (C(r, /₀))~"\ and the result then follows from (10.8.4), (5.7.5), and (8.3.2). The knowledge of C(s₉1) enables one to give the explicit solution of (10.6.2) taking the value JC_Q for t = t₀: (10.8.6) The function ii(0 = C(t, t₀) • x₀ + \(C(t, s) • b(s)) ds *Jto*